analytic integration of singular kernels for boundary

Composites Part B, 108, 393412 (2017).

Analytic integration of singular kernels for

boundary element analysis of plane orthotropic

media

E. J. BarberoWest Virginia University, Morgantown USA

and

F. Vetere, A. Madeo 1, and R. ZinnoUniversity of Calabria, Arcavacata di Rende, CS, Italy

Abstract

Both composite materials and stress concentration are common issues in modern structural engi-neering. In this paper, analytical integration of the singular kernels is performed for boundaryelement analysis (BEM) of elastic, plane orthotropic media with stress concentrations. Analyticalintegration leads to accuracy and eciency improvements over FEM. Furthermore, high continuity(HC), quadratic spline interpolation on the boundary is used to further improve accuracy at lowcomputational cost when compared to FEM. The advantages of BEM for calculation of displace-ments and stresses near stress raisers in orthotropic plates are shown. Particular attention is paidto ecient interpolation for approximating boundary quantities and to precision of computation forevaluating boundary integrals. Such improvements lead to accurate computation of both displace-ments and stresses in both the boundary and the domain. Thus, the advantages of the proposedmethod are accuracy and low computational cost.

Keywords

Boundary Element Method (BEM), B. Elasticity, B. Stress concentrations, C. Computational mo-deling.

1 Introduction

Orthotropic materials, including laminated composite materials, are extensively used in modernindustry. The theory of elasticity for orthotropic bodies is well established and solutions have beenobtained for simple problems [1,2]. The presence of cracks increases the complexity of the analysis[37]. Therefore, complex problems of orthotropic, elastic bodies are analyzed with numericalmethods, such as the Finite Element Method (FEM) [813] and the Boundary Element Method(BEM) [1427]. Application examples of the BEM can be found, for example in [2833]. Specic

1Corresponding author. The nal publication is available at http://dx.doi.org/10.1016/j.compositesb.2016.10.021

1

http://dx.doi.org/10.1016/j.compositesb.2016.10.021

http://dx.doi.org/10.1016/j.compositesb.2016.10.021

Composites Part B, 108, 393412 (2017). 2

application of BEM to plasticity and fracture can be found, for example in [3437]. BEM analysisof interphase cracks and transverse isotropy in bimaterials is presented in [38,39].

The boundary element approach can be used to construct ecient algorithms for numericalanalysis of engineering problems. Some characteristics of this method are very attractive, namelythe reduction of the dimension (e.g., from 2D to 1D) of the discrete model in comparison withdomain methods (e.g., FEM), and the mixed nature (tractiondisplacement) of the method, whichyields comparable accuracy in both displacements and tractions.

Green [40] rst introduced the fundamental solutions for 2-D orthotropic bodies under a concen-trated force. Rizzo and Shippy (1970) [41] introduced the fundamental solutions into the boundaryintegral equations for numerical elastic analysis of stress concentration. Recently, the fundamentalsolutions for orthotopic plane problems were improved [4247], but numerical integration was usedto compute the boundary coecients.

Analytical evaluation of the boundary coecients has been used for 2-D isotropic, plane andbending elastic problems [4850] and in BEM for orthotropic, plane, potential problems [51,52]. Inthis paper, analytical integration is developed for the 2D orthotropic problem.

In the context of the boundary element method for plane orthotropic media, the aim of this workis to provide an accurate evaluation of the stress eld with low computational cost. Accuracy andeciency are achieved by rening both the boundary interpolation and the integration process. Insummary, the boundary is discretized into macro-elements using a quadratic high continuity (HC)spline approximation to ensure C1 continuity using few control points. Then, analytical integrationof coecients is carried out on linear piecewise boundaries. The exact evaluation of integrals isdecisive for an accurate, yet inexpensive computation of the domain stress eld from the boundarysolution. Finally, and assessment of the performance of the proposed methodology is presented.

2 Orthotropic plane problem

Consider a 2D orthotropic elastic body described in a rectangular Cartesian system x1, x2. Thebehavior of the body is described by two-dimensional elds of displacement, stress, and strain. Theproblem is governed by the customary equations, namely kinematic, constitutive, and equilibriumequations (e.g., [8, Chapter 2]).

2.1 Fundamental Solution

Consider an elastic orthotropic problem on a 2D innite domain subjected to a concentrated forcef? applied at the source point ξ. The fundamental solution for this problem is obtained by rstrewriting the equilibrium equations, taking into account of the Hooke's law, into form

Lu + b = 0 (1)

where L is the dierential operator

[L]

=

C11∂21 + C66∂

22 (C11 + C66) ∂1∂2

(C11 + C66) ∂1∂2 C22∂22 + C66∂

21

(2)

and u is the displacement eld, b are the body forces, ∂i = ∂/∂xi, and Cij are the coecients ofthe stiness tensor in Voigt contracted notation. Solving the system (2), the following fundamentalsolution was obtained by Huang [42], for displacements


u∗11 (ξ, x) = D[√λ1A

22 ln z1 −

√λ2A

21 ln z2

]u∗12 (ξ, x) = DA1A2

[arctan

(r2√λ2r1

)− arctan

(r2√λ1r1

)]u∗21 (ξ, x) = u12 (ξ, x)

u∗22 (ξ, x) = −D[A2

1√λ1

ln z1 −A2

2√λ2

ln z2

](3)

where the generic term u∗αi (ξ, x) of the fundamental solution represents the component of thedisplacement at the eld point x in xi direction due to the application at the source point ξ of aunit force directed along xα.

The fundamental solution for the tractions is

t∗11 (ξ, x) = D

[√λ2A1

z22

−√λ1A2

z21

](r1n1 + r2n2)

t∗12 (ξ, x) = D

(√λ1A1

z21

−√λ2A2

z22

)r1n2 −

(√λ1

λ1

A1

z21

−√λ2

λ2

A2

z22

)r2n1

t∗21 (ξ, x) = D

(λ1

√λ1A2

z21

− λ2

√λ2A1

z22

)r1n2 −

(√λ1A2

z21

−√λ2A1

z22

)r2n1

t∗22 (ξ, x) = D

[√λ1A1

z21

−√λ2A2

z22

](r1n1 + r2n2)

(4)

where nk represent the components of the unit normal at the eld point x, and the generic termt∗αi (ξ, x) of the fundamental solution represents the component of the traction at the eld point xin xi direction due to the application at the source point ξ of a unit force directed along xα.

The distance between the source ξ and the eld point x is

rk (ξ, x) = xk − ξk (5)

where

λ1 + λ2 =2S12 + S66

S22(6)

λ1λ2 =S11

S22(7)

Ak = S11 − λkS22 (8)

z2k = λkr

21 + r2

2 (9)

D =1

2π (λ1 − λ2)S22(10)

Equations (6) and (7) imply that


λ1,2 =2S21 + S66

2S22±

√(2S21 + S66

2S22

)2

−(S11

S22

)(11)

where Sij are the components of the compliance tensor in Voigt contracted notation.

2.2 Integral equation on the boundary

The integral equations on the boundary can be derived using Betti's reciprocal work theorem, orthe weighted residuals method, resulting in Somigliana's equation [53]

u (ξ) =

∫Γu∗

T(ξ, x) t (x)dΓ−

∫Γt∗

T(ξ, x)u (x)dΓ +

∫Ωu∗

T(ξ, x)b (x)dΩ (ξ ∈ Γ) (12)

When evaluating (12) on the boundary, singularities of order O(1/r2) require evaluation theintegral as Cauchy Principal Values [54]

u (ξ) =

∫Γu∗

T(ξ, x) t (x)dΓ−

(MT

c u(ξ) +

∫Γt∗

T(ξ, x)u (x)dΓ

)+

∫Ωu∗

T(ξ, x)b (x)dΩ (ξ ∈ Γ)

(13)which can be written as

cTu (ξ) =

∫Γu∗

T(ξ, x) t (x) dΓ −

∫Γt∗

T(ξ, x)u (x) dΓ +

∫Ωu∗

T(ξ, x)b (x) dΩ (ξ ∈ Γ) (14)

where b are the body forces, u∗ and t∗ are the fundamental solutions (3) and (4), Ω is the 2Ddomain, Γ is the boundary of the domain, and c is 2 × 2 coecient matrix which depends on thesurface geometry and it is dened as following

c = I + MTc =

1 0

0 1

+

c11 0

0 c22

(15)

where c11 and c22 are evaluated in the Appendix (4).

2.3 Discrete model

The approximate solution requires discretization of both the geometry of the contour and the me-

chanical variables, i.e., displacement and traction. The geometry of the contour can be representedthrough a system of curvilinear coordinates or through piecewise linearization. The curvilinearrepresentation is the most general, but requires that the boundary integrals be evaluated in nume-rically. Numerical integration requires us to pay particular attention to the calculation of singularand nearly singular integrals. On the other hand, a piecewise linear representation of the contourallows us to use analytical integration for both, the solution on the boundary Γ and the solutioninside the domain Ω.

For polygonal domains, the discretization into boundary elements proceeds as follows. The dis-continuities of the boundary (e.g., corners) and the boundary conditions (e.g., attachement points)are used to subdivide the boundary into macro-elements. Macro-elements (denoted by Mi in Figure1) are used to discretize the boundary; not the domain. In this way, the integral equations are


Figure 1: Discretization with macro-elements Mi, subdivided into elements Γe. Note that each change froma displacement boundary condition to a traction one, or viceversa, requires a new macro-element.

represented through summations extended over the number of macro-elements. Furthermore, thediscretization of the mechanical variables on the boundary requires that each macro-element besubdivided into a nite number of evenly spaced elements (denoted by Γe in Figure 1).

The boundary variables are approximated by shape functions φe(ζ) (Figure 2) as follows

u (ζ) =

ne∑e=1

ue (ζ) =

ne∑e=1

φue ue; φue =φ(j)(ζ), φ(j+1)(ζ), φ(j+2)(ζ)

t (ζ) =

ne∑e=1

te (ζ) =

ne∑e=1

φte ue; φte =φ(j)(ζ), φ(j+1)(ζ), φ(j+2)(ζ)

(16)

2.3.1 High Continuity (HC) Interpolation

The quality of the numerical solution depends on the quality of the representation of the variableson the boundary. An interpolation that has been shown to provide high quality interpolation isthe High Continuity (HC) interpolation that was proposed by Aristodemo [55] and used for BEMin [48,49].

In HC interpolation, the boundary variables are represented by a quadratic B-spline approxi-mation which guarantees C1 continuity. In each element, the shape function is constructed usingthree control points (see Fig. 3(a)), associated with one node placed at the midpoint of the elementitself, and one node on each of the two adjacent elements.

Each macro-element is divided into n elements. For n elements, while a piecewise constantinterpolation uses n parameters, the HC interpolation HC uses n + 2. To enforce C1 continuity,while Hermite interpolation uses 2n parameters, HC uses only n+ 2, achieving the same continuity.

For n + 2 parameters, HC interpolation requires n + 2 sources located on the macro-element.The sources are located at the midpoint of the internal elements, plus one additional source foreach of the end elements. The sources in the end elements are located as shown in Fig. (4), whereα = 0.6a and β = 1.4a, where a is the element half-length. The values of α, β, are chosen bynumerical optimization [49].


Figure 2: Quadratic interpolation of displacements.


(a) (b)

(c)

Figure 3: Shape functions and arrangement of the interpolation parameters.


Figure 4: Arrangement of the sorce points on a macro-element.

In (14), the displacements and the tractions eld u and t can be approximated by the followinggeneric function

fi (ζ) =

3∑k=1

φ(k) (ζ)f(k)i =

3∑h=1

(2∑

h=0

chk(ζh)

)f

(k)i ; i = 1 . . . n+ 2 (17)

where ζ is the local element coordinate, with −1 < ζ < 1 (Figure 3.a), φ(k) (ζ) is the interpolationfunction associated to the function fi at node k, and chk is the coecient of degree h of thepolynomial corresponding to nodal parameter k. Also, n is the number of elements on a macro-element. For example, if there are n = 8 elements (Figure 4), then we need 10 HC parameters.

The general expression of these functions is [55]

φ(1) (ζ) =1

4 (s+ 1)

(1− 2ζ + ζ2

)φ(2) (ζ) =

1

4 (s+ 1) (d+ 1)

((2 + 3 (s+ d) + 4sd) + 2 (d− s) ζ − (d+ s+ 2) ζ2

)φ(3) (ζ) =

1

4 (d+ 1)

(1 + 2ζ + ζ2

)(18)

where s = 1, d = 0, for the leftmost element in the macro-element, s = 0, d = 1, for the rightmostelement, and s = 1, d = 1, for inside elements. In this work, the shape functions are generatedusing s = d = 1 (inside element). Use of all three types of HC elements is onerous and it does notprovide any advantage for the work carried out in this particular case, i.e., 2D plane orthotropicmedia. Therefore, we have

φ(1) (ζ) =1

8− 1

4ζ +

1

8ζ2

φ(2) (ζ) =3

4− 1

4ζ2

φ(3) (ζ) =1

8+

1

4ζ +

1

8ζ2

(19)


2.4 Determination of the solution on the boundary

The discrete form of (14) allows to construct an algebraic system of equations in terms of contourvariables. Using HC interpolation, the displacement and traction elds are represented inside eachelement by the following relations

u (x) = φueue

t (x) = φtete(20)

with

φe =[φ(j) φ(j+1) φ(j+2)

]uTe =

u(j)1 u

(j+1)1 u

(j+2)1

u(j)2 u

(j+1)2 u

(j+2)2

tTe =

t(j)1 t(j+1)1 t

(j+2)1

t(j)2 t

(j+1)2 t

(j+2)2

(21)

where the generic terms φe representing φue,φte, are given by (19). In (21), the subscripts 1, 2,represent the eld directions, and the superscripts j, j + 1, j + 2 represent one of the three nodesinvolved in the interpolation for element e. Therefore, the discrete form of (14) becomes

cTu (ξ) +

ne∑e=1

∫Γe

t∗T

(ξ, x)φueue dΓe =

=

ne∑e=1

∫Γe

u∗T

(ξ, x)φtete dΓe +

nc∑c=1

∫Ωc

u∗T

(ξ, x)b (x) dΩc (ξ ∈ Γ) (22)

where ξ denotes the position of the source, ne the number of elements along the contour, and ncis the number of cells in the domain. On the contour, arranging a number of sources equal to thenumber of parameters in the discretization, (22) becomes∑

e

Heue =∑e

Gete + b (23)

The contributions of element e to the matrices H and G are dened by

He =

∫Γe

t∗T

(ξ, x)φue dΓe

Ge =

∫Γe

u∗T

(ξ, x)φte dΓe

(24)

Using (23)(24), the system (22) can be rewritten as follows

Hu = Gt + b (25)

Then, taking in account the boundary conditions of the problem, the system (25) can be rewrit-ten in the compact form as

Ax = f (26)


where the coecient matrix A contains both the coecients of H, G. The vector of unknowns xcollects the unknown values of the the displacement eld u and tractions t. The vector f is composedby the terms resulting from the product of the known values of u and t by the correspondingcoecients in the matrices H and G, plus the body forces b, as it is explained next.

The boundary is composed of a collection of macro-elements that are subjected to either specieddisplacement or specied traction. For those macro-elements under specied displacement, thetractions are unknown, so in (26), A receives values from G, the unknowns x are the tractions t,and the RHS is calculated as f = Hu− b. For the remaining macro-elements, which are subjectedto specied traction, the displacements are unknown, so A receives values from H, the unknownsx are the displacements u, and the RHS is calculated as f = Gt + b.

The size of matrixA is very small when compared to the matrix of domain discretization methods(e.g., FEM), but it is full and not symmetric. Since x includes both u and t, BEM is said to bemixed ; that is, both displacements and tractions are evaluated simultaneously with comparableprecision.

2.5 Analytical integration of boundary coecients

The integrands in the boundary integrals (22) involve the products between the shape functions andthe fundamental solutions (3) and (4). Analytical computation of these integrals is convenientlyperformed using a local coordinate system centered at the midpoint of the boundary element. Theintegrals have the following typical form∫

Γf∗φ(k) (x) dΓ = a

2∑h=0

chk

∫ 1

−1f∗i ζ

(h)dζ ; i = 1 . . . n+ 2 (27)

where the abscissa ζ = x/a is taken in a local system centered on the eld element, and a indicatesthe half-length of the eld element (Figure (5)). Note that i = 1 . . . n+2, where n+2 is the numberof parameters on a macro-element. For example, if there are n = 8 elements (Figure 4), then wehave 10 HC parameters.

Figure 5: Coordinates of the source point S(x, y) in the local system placed on the eld element.


The integrals (27) have the following typical forms

E(k)(h)j =

∫ ζ2

ζ1

ζh

(z2k)j

dζ

G(k)(h)j =

∫ ζ2

ζ1

ln (zk)(z2k)jζhdζ

A(k)(h)j =

∫ ζ2

ζ1

arctan

(y√

λk (ζ − x)

)(z2k)jζhdζ

(28)

where x, y are the coordinates of the source point expressed in the local system of the eld element

and j is the power of the z2k term. The analytical solutions of the indenite integrals E(k)

(h)j can

be written in the following recursive form

E(k)(h)j =

1

2j − h− 1

[ζh−1(z2k

)j−1

]ζ2ζ1

+ 2x (j − h)E(k)(h−1)j +

+ (h− 1)[(n1 + n2λk) x

2 + (n1λk + n2) y2]E(k)

(h−2)j

(29)

When h = 2j − 1, it is necessary to use the following equation

E(k)(h)j = E(k)

(h−2)j−1 −

[(n1 + n2λk) x

2 + (n1λk + n2) y2]E(k)

(h−2)j + 2xE(k)

(h−1)j (30)

and, to initialize the recursive process when y 6= 0, the following integrals are used

E(k)(0)1 =

1√λky

[arctan

((n1 + n2

√λk)

(ζ − x)(n1

√λk + n2

)y

)]ζ2ζ1

(31a)

E(k)(1)1 =

1(n1 + n2

√λk)[ln (zk)

]ζ2ζ1

+ xE(k)(0)1 (31b)

E(k)(h)0 =

[ζh+1

h+ 1

]ζ2ζ1

(31c)

E(k)(0)j+1 =

1

2 (n1λk + n2) y2j

[

(ζ − x)(z2k

)j]ζ2ζ1

+ (2j − 1)E(k)(0)j

(31d)

For y = 0 some of these expressions degenerate, and must be replaced by the following

E(k)(0)j =

[1

(ζ − x)2j−1

]ζ2ζ1

1

(n1 + n2λk) (1− 2j)(32)

Integrals of type G(k)(h)j can be represented in closed form. For the problem of 2D plane

orthotropic media, only evaluation of integrals G(k)(h)0 is required, which for y 6= 0 become

G(k)(h)0 =

1

h+ 1

[ln (zk) ζ

h+1

]ζ2ζ1

− (n1 + n2λk)(E(k)

(h+2)1 − xE(k)

(h+1)1

) (33)


When y = 0, (33) becomes

G(k)(h)0 =

1

h+ 1

[ζh(

(ζ − x) ln (zk)−ζ

h+ 1

)]ζ2ζ1

+ hxG(k)(h−1)j

(34)

and the rst integral of the recursive process is

G(k)(0)0 =

[(ζ − x) ln (zk)− ζ

]ζ2ζ1

(35)

Also integrals of type A(k)(h)j can be represented in closed form. For the problem of 2D plane

orthotropic media, only integrals A(k)(h)0 are required

A(k)(h)0 =

=

[− h

h+ 1

(xy(h−1)

√λk

)+

y

2√λkln (zk) ζ

(h) +(h− 1)

6

y3√λ3k

ln (zk) +

+h

2

y2

λkarctan

(√λk (ζ − x)

y

)x(h−1) − arctan

(y

λk (ζ − x)

)(ζ(h+1) − x(h+1)

h+ 1

)]ζ2ζ1

(36)

and the rst integral of the recursive process is

A(k)(0)0 =

[y

2√λkln (zk)− arctan

(y

λk (ζ − x)

)(ζ − x)

]ζ2ζ1

(37)

Equations (27) to (37) allow us to evaluate the integrals in (22). After integration, the integrals∫Γe

u∗Tφte dΓe and∫

Γet∗Tφte dΓe are expressed as combinations of E(k)

(h)j , A(k)

(h)j , and G(k)

(h)j ,

which are called U(h)ij , and T

(h)ij , and reported in the Appendix. The expressions of U

(h)ij , T

(h)ij ,

conveniently assembled, allow us to evaluate H and G in (25).

2.6 Domain solution

The solution in the interior of the domain can be obtained from the solution on the boundary.When the source point and the eld point are very close (rk → 0), the integrals of the form∫

Γ f∗T (ξ, x) g (x) dΓ become singular, so they are evaluated using Cauchy Principal Values (CPV)

(see the Appendix (4)). Since there are no singularities in the domain, c = I,2 and using (14) weobtain

u (ξ) =

∫Γu∗

T(ξ, x) t (x) dΓ −

∫Γt∗

T(ξ, x)u (x) dΓ +

∫Ωu∗

T(ξ, x)b (x) dΩ (ξ ∈ Ω) (38)

and in discrete form

u (ξ) =

=

ne∑e=1

∫Γe

u∗T

(ξ, x)φtete dΓe −ne∑e=1

∫Γe

t∗T

(ξ, x)φueue dΓe+

nc∑c=1

∫Ωc

u∗T

(ξ, x)b (x) dΩc (ξ ∈ Ω)

(39)

2I : Identity matrix.


Calculation of stress elds requires dierentiation of the displacement eld to obtain the straineld, as dictated by the kinematic equations [8, (1.4)], and further use of the constitutive equations[8, (1.55)], resulting in

σ (ξ) =

∫ΓD (ξ, x) t (x) dΓ −

∫ΓS (ξ, x)u (x) dΓ +

∫ΩD (ξ, x)b (x) dΩ (ξ ∈ Ω) (40)

and in discrete form

σ (ξ) =

ne∑e=1

∫Γe

Dφtete dΓe−ne∑e=1

∫Γe

Sφueue dΓe+

nc∑c=1

∫Ωc

Dbc (x) dΩc (ξ ∈ Ω) (41)

where for 2D media

DT =

[D111 D121 D211 D221

D112 D122 D212 D222

]; ST =

[S111 S121 S211 S221

S112 S122 S212 S222

](42)

and by analytic integration we obtain

D111 = D

(√λ2A1

r1

z22

−√λ1A2

r1

z21

)D122 = D

(λ1

√λ1A2

r1

z21

− λ2

√λ2A1

r1

z22

)D121 = D

(√λ2A1

r2

z22

−√λ1A2

r2

z21

)D112 = D121

D211 = D

(A2√λ2

r2

z22

− A1√λ1

r2

z21

)D222 = D

(√λ1A1

r2

z21

−√λ2A2

r2

z22

)D212 = D

(√λ1A1

r1

z21

−√λ2A2

r1

z22

)D212 = D221

(43)

and

S111 = D

[1√λ2z2

2

− 1√λ1z2

1

− 2

(√λ2r

21

z42

−√λ1r

21

z41

)]n1 − 2

[√λ2r1r2

z42

−√λ1r1r2

z41

]n2

S112 = D

−2

[√λ2r1r2

z42

−√λ1r1r2

z41

]n1 +

[√λ2

z22

−√λ1

z21

− 2

(√λ2r

22

z42

−√λ1r

22

z41

)]n2

S121 = S211 = S112

S122 = D

[−√λ2

z22

+

√λ1

z21

+ 2

(λ2

√λ2r

21

z42

− λ1

√λ1r

21

z41

)]n1 + 2

[λ2

√λ2r1r2

z42

− λ1

√λ1r1r2

z41

]n2

S212 = S221 = S122

S222 = D

2

[λ2

√λ2r1r2

z42

− λ1

√λ1r1r2

z41

]n1 +

[−λ2

√λ2

z22

+λ1

√λ1

z21

+ 2

(λ2

√λ2r

22

z42

− λ1

√λ1r

22

z41

)]n2

(44)

Integral terms of the type Dijl and Sijl, written as functions of recurring integrals, are given inthe Appendix.


The body force b has been included for completeness, but no provisions have been made for itsintegration because we do not plan to use for our intended application (see Conclusions). Body forceintegration is a classical topic, where the goal is to avoid domain integration and use only boundaryintegration to guarantee all the advantages of BEM. Some classical papers describing techniques fortransforming BEM domain integrals to the boundary are noted here [5662].

3 Numerical Results

A number of cases are presented here to demonstrate the application of the proposed methodology toelastic analysis of 2-D orthotropic medium by the analytical integration of the kernels. The principalmaterial directions are aligned with the Cartesian coordinate directions. All Abaqus simulationswere performed using a uniform mesh of S8R elements. All elements had the same square shape.The reference values (Uref or σref ) used in the graphs have been evaluated numerically using a veryne mesh (i.e. values at convergence ) employing Abaqus. A pointwise measure of the error hasbeen used. The log(|.|) has been introduced to better emphasize the rate of convergence, where ||denotes absolute value.

Please note that in the tabulated results one cannot measure the error by comparing BEM andFEM results between any pair of BEM and FEM discretizations, not even between the ner BEMand ner FEM meshes, because BEM and FEM meshes are not comparable. While a FEM mesh isa discretization of the domain, a BEM mesh is a discretization of the boundary. Therefore, identicalresults cannot be obtained. Only convergence and rate of convergence are meaningful, both of whichare satised for both BEM and FEM, in all examples that follow.

3.1 Square plate under uniform load

Consider an orthotropic square plate with side L = 100 mm, subjected to uniformly distributedload qy = 1.0 N/mm along the principal material direction that coincides with the y-coordinate.Material properties are Ey = 161 MPa, Ex = 90.27 MPa, Gxy = 7.17 MPa, νxy = 0.28 and thicknesst = 1.2 mm. The plate is clamped at x = 0 loaded with qy at y = L (Figure 6).

The results are compared with numerical results obtained using Abaqus. Comparison of thedisplacements at points A=(L, 0), B=(L,L), C=(L/3, 2/3L) and D=(2/3L,L/3) are presented inTables 1 and 2. Both tables refer to the same degrees of freedom (dof) shown in Table 1.

Convergence of displacement at boundary points located in point A and B vs. number of degreesof freedom (dof) is depicted in Figure 7. It can be seen that BEM converges monotonically to thedisplacements using less dof than Abaqus. Convergence of displacement at domain point C vs.number of degrees of freedom (dof) is shown in Figure 8. It can be seen that BEM converges to thedisplacement using less dof than Abaqus.

Comparison of traction tn at point E=(0, L/2) is shown in Table 3. Comparison of stresses atpoints C is presented in Table 4. The table refers to the same dof shown in Table 1. Convergenceof stress at domain point C vs. number of degrees of freedom (dof) is shown in Figure 9. The BEMcode shows a faster, monotonic convergence compared with Abaqus.


Figure 6: Square plate under uniform load.

Table 1: Comparison of displacements at points A and B, in [mm].

BEM Abaqus

N totel L/Nel dof UAx UBy N tot

el L/Nel dof UAx UBy

12 33.333 40 1.2216 8.3447 9 33.333 120 1.1671 6.962024 16.666 64 1.1908 8.2409 36 16.666 399 1.1447 7.607336 11.111 88 1.1895 8.2281 81 11.111 840 1.1507 7.824048 8.333 112 1.1773 8.2261 144 8.333 1443 1.1539 7.927960 6.666 136 1.1718 8.2258 225 6.666 2208 1.1557 7.989272 5.555 160 1.1690 8.2257 324 5.555 3135 1.1568 8.029684 4.761 184 1.1674 8.2257 441 4.761 4224 1.1575 8.058396 4.166 208 1.1665 8.2256 576 4.166 5475 1.1580 8.0796108 3.703 232 1.1660 8.2254 729 3.703 6888 1.1584 8.0962120 3.333 256 1.1657 8.2253 900 3.333 8463 1.1587 8.1094


Table 3: Convergence of normal traction tn at point E.

N totel BEM tn N tot

el Abaqus tn12 0.012605 4 0.00872820 0.019614 16 0.017994132 0.019092 64 0.020206220 0.019167 256 0.020206308 0.019195 1024 0.019763396 0.019208 4096 0.019122484 0.019216 16384 0.019126

Table 2: Comparison of displacements at points C and D, in [mm].

BEM Abaqus

UCx UCy UDx UDy UCx UCy UDx UDy

−0.1353 3.8428 0.1546 6.5786 −0.0244 2.1964 0.1167 5.2396−0.1234 3.7846 0.1410 6.4863 −0.1158 3.2045 0.1373 5.8530−0.1220 3.7726 0.1388 6.5539 −0.1150 3.3630 0.1377 6.0612−0.1214 3.7680 0.1380 6.4607 −0.1167 3.4707 0.1373 6.1623−0.1211 3.7657 0.1376 6.4575 −0.1179 3.5285 0.1371 6.2225−0.1209 3.7644 0.1374 6.4556 −0.1184 3.5678 0.1370 6.2622−0.1208 3.7636 0.1373 6.4545 −0.1189 3.5955 0.1369 6.2904−0.1207 3.7631 0.1372 6.4537 −0.1192 3.6162 0.1369 6.3115−0.1206 3.7627 0.1371 6.4532 −0.1194 3.6323 0.1368 6.3279−0.1206 3.7624 0.1371 6.4528 −0.1196 3.6451 0.1368 6.3409

-4

-3

-2

-1

1 2 3 4

log(|

1-U

/Ure

f|)

log(dof)

UxA

BEM,Ux

A Abaqus

(a) UAx

-5

-4

-3

-2

-1

0

1 2 3 4

log(|

1-U

/Ure

f|)

log(dof)

UyB

BEM,Uy

B Abaqus

(b) UBy

Figure 7: Convergence of displacement at boundary points A and B vs. degrees of freedom (dof).


Table 4: Comparison of stresses at point C, in [MPa].

BEM Abaqus

σCxx σCyy τCxy σCxx σCyy τCxy

0.54971 −0.67136 −0.25424 0.30868 −0.93328 −0.129800.54617 −0.66513 −0.24929 0.60545 −0.60281 −0.181170.54546 −0.66450 −0.24457 0.53595 −0.68128 −0.236430.54515 −0.66417 −0.24264 0.54351 −0.66386 −0.235950.54508 −0.66413 −0.24225 0.54446 −0.66640 −0.237560.54504 −0.66411 −0.24200 0.54368 −0.66524 −0.238860.54502 −0.66409 −0.24183 0.54388 −0.66528 −0.239430.54500 −0.66409 −0.24170 0.54388 −0.66510 −0.239840.54500 −0.66409 −0.24161 0.54390 −0.66504 −0.240130.54498 −0.66408 −0.24150 0.54393 −0.66498 −0.24034

-3

-2

-1

0

1 2 3 4

log(|

1-U

/Ure

f|)

log(dof)

UxC

BEM,Ux

C Abaqus

(a) UCx

-4

-3

-2

-1

0

1 2 3 4

log(|

1-U

/Ure

f|)

log(dof)

UyC

BEM,Uy

C Abaqus

(b) UCy

Figure 8: Convergence of displacement at domain point C vs. degrees of freedom (dof).


-5

-4

-3

-2

-1

0

1 2 3 4

log(|

1-σ

/σre

f|)

log(dof)

σxxC

BEM,σxx

C Abaqus

(a) σCxx

-4

-3

-2

-1

0

1 2 3 4

log(|

1-σ

/σre

f|)

log(dof)

σyyC

BEM,σyy

C Abaqus

(b) σCyy

-6

-5

-4

-3

-2

-1

0

1 2 3 4

log(|

1-σ

/σre

f|)

log(dof)

τxyC

BEM,τxy

C Abaqus

(c) τCxy

Figure 9: Convergence of stress at domain point C vs. degrees of freedom (dof).


3.2 Cantilever plate under uniform shear load

Consider an orthotropic cantilever plate with base b = 200 mm (along x) and height h = 10 mm(alogn y), subjected to uniformly distributed shear load qy = −0.030 N/mm at x = b. The plateis clamped at x = 0. The principal material direction coincides with the x-coordinate. Materialproperties are Ex = 85 MPa, Ey = 74 MPa, Gxy = 10 MPa, νxy = 0.3 and thickness t = 1 mm. Foroptimum accuracy, the length of the elements are constant for all macro-elements.

A typical mesh is shown in Figure 10. The element size is shown as h/Nel in Table 5.

Figure 10: Typical mesh used for Abaqus discretization. Cantilever plate under uniform shear load.

The results are compared with numerical results obtained using Abaqus at points A=(b,0),B=(b,h), C=(0, h/2), and D=(b/5, h/2). Comparison of displacements at points A and B arepresented in Table 5. Convergence of displacement at boundary points A and B vs. number ofdegrees of freedom (dof) is shown in Figure 11. It can be seen that BEM shows an higher rate ofconvergence than Abaqus.

Convergence of stress at domain point D vs. degrees of freedom (dof) is reported in Table 6and Figure 12, while tangential traction tt at point C in Table 7 and Figure 12. It can be seen thatBEM converges to the traction/stress using less dof that Abaqus.


BEM Abaqus

N totel h/Nel dof UAy UBx N tot

el h/Nel dof UAy UBx

42 10 100 −111.2293 4.1389 20 10 309 −113.1230 4.225284 5 184 −111.3818 4.1535 80 5 975 −113.2740 4.2290168 2.5 352 −112.1102 4.1825 320 2.5 3387 −113.4370 4.2333336 1.25 688 −112.8399 4.2085 1280 1.25 12531 −113.5280 4.2358672 0.625 1360 −113.1830 4.2206 5120 0.625 48099 −113.6040 4.23781344 0.3125 2704 −113.3130 4.2252


Table 6: Comparison of stress τxy [MPa] at point D.

BEM Abaqus

N totel h/Nel dof τDxy N tot

el h/Nel dof τDxy

42 10 100 −0.04490 20 10 309 −0.0292684 5 184 −0.04400 80 5 975 −0.05265168 2.5 352 −0.04438 320 2.5 3387 −0.04687336 1.25 688 −0.04493 1280 1.25 12531 −0.04546672 0.625 1360 −0.04520 5120 0.625 48099 −0.045111344 0.3125 2704 −0.04530

Table 7: Comparison of tangential traction tt at point C.

BEM Abaqus

N totel h/Nel dof tCt N tot

el h/Nel dof tCt

42 10 100 0.4739 20 10 309 0.0194126 3.333 268 0.1245 80 5 975 0.0105210 2 436 0.1237 320 2.5 3387 0.0043462 0.909 940 0.1125 1280 1.25 12531 0.0022714 0.588 1444 0.1112 5120 0.625 48099 0.10471470 0.285 2956 0.1105

-4

-3

-2

-1

2 3 4 5

log(|

1-U

/Ure

f|)

log(dof)

UyA

BEM,Uy

A Abaqus

(a) UAy

-4

-3

-2

-1

2 3 4 5

log(|

1-U

/Ure

f|)

log(dof)

UxB

BEM,Ux

B Abaqus

(b) UBx

Figure 11: Convergence of displacement at boundary points A and B vs. number of degrees of freedom (dof).


-4

-3

-2

-1

0

1

2 3 4 5

log(|

1-t

/tre

f|)

log(dof)

ttC

BEM,tt

C Abaqus

(a) tCt

-4

-3

-2

-1

0

2 3 4 5

log(|

1-σ

/σre

f|)

log(dof)

τxyD

BEM,τxy

D Abaqus

(b) τDxy

Figure 12: Convergence of tangential traction at point C and stress at domain point D vs. number of degreesof freedom (dof).

3.3 L-shape plate under uniform shear load

Consider an orthotropic irregular plate subjected to a uniformly distributed shear load qy = −0.0135N/mm along the principal material direction that coincides with the y-coordinate (Figure 13).Material properties are Ey = 85 MPa, Ex = 74 MPa, Gxy = 20 MPa, νxy = 0.3. The length isL = 100 mm and thickness t = 1.3 mm. Element size is uniform along the boundary, and given asL/Nel in Table 8 for each level of discretization.

The results are compared at four points. The rst three points, A=(2.5L, 1.5L), B=(2.5L, 2.5L)and C=(L,L) are located on the boundary. The point D=(1.5L, 2.0L) is located in the domain.

Convergence of displacement at boundary points located in point A and B vs. number of degreesof freedom (dof) is shown in Table 8 and Figure 14. It can be seen that BEM converges monotonicallyto the displacement solution using less dof than Abaqus. Convergence of displacement at point Cvs. number of degrees of freedom (dof) is shown in Table 9and Figure 15. It can be seen thatconverges to the displacement using less dof that Abaqus.

Convergence of stress at domain point D vs. number of degrees of freedom (dof) is reportedin Table 10 and Figure 16. The BEM code shows faster, monotonic convergence compared withAbaqus.


Figure 13: L-shape plate under uniform shear load.


BEM Abaqus

N totel L/Nel dof UBx UAy N tot

el L/Nel dof UBx UAy

20 50 64 1.0646 −1.5967 16 50 207 1.0382 −1.491240 25 104 1.0777 −1.6052 64 25 699 1.0601 −1.547580 12.5 184 1.0801 −1.6032 256 12.5 2547 1.0709 −1.5748160 6.25 344 1.0807 −1.6014 1024 6.25 9699 1.0759 −1.5873320 3.125 664 1.0808 −1.6003 4096 3.125 37827 1.0781 −1.5931

Table 9: Comparison of displacements at point C, in [mm].

BEM Abaqus

N totel L/Nel dof UCx UCy N tot

el L/Nel dof UCx UCy

20 50 64 0.2752 −0.2822 16 50 207 0.3241 −0.259840 25 104 0.3155 −0.2381 64 25 699 0.3167 −0.260580 12.5 184 0.3112 −0.2419 256 12.5 2547 0.3113 −0.2593160 6.25 344 0.3078 −0.2446 1024 6.25 9699 0.3075 −0.2574320 3.125 664 0.3052 −0.2465 4096 3.125 37827 0.3048 −0.2557


-4

-3

-2

-1

1 2 3 4 5

log(|

1-U

/Ure

f|)

log(dof)

UyA

BEM,Uy

A Abaqus

(a) UAy

-5

-4

-3

-2

-1

1 2 3 4 5

log(|

1-U

/Ure

f|)

log(dof)

UxB

BEM,Ux

B Abaqus

(b) UBx


-3

-2

-1

1 2 3 4 5

log(|

1-U

/Ure

f|)

log(dof)

UxC

BEM,Ux

C Abaqus

(a) UCx

-3

-2

-1

0

1 2 3 4 5

log(|

1-U

/Ure

f|)

log(dof)

UyC

BEM,Uy

C Abaqus

(b) UCy

Figure 15: Convergence of displacement at boundary point C vs. number of degrees of freedom (dof).

Table 10: Comparison of stress at point D, in [MPa].

BEM Abaqus

N totel L/Nel dof σDxx τDxy N tot

el L/Nel dof σDxx τDxy

20 50 64 −0.00058 0.01520 16 50 207 −0.00786 0.0224740 25 104 −0.00122 0.01528 64 25 699 −0.00185 0.0156380 12.5 184 −0.00137 0.01524 256 12.5 2547 −0.00171 0.01525160 6.25 344 −0.00143 0.01522 1024 6.25 9699 −0.00160 0.01519320 3.125 664 −0.00146 0.01521 4096 3.125 37827 −0.00155 0.01518


-3

-2

-1

0

1

1 2 3 4 5

log(|

1-σ

/σre

f|)

log(dof)

σxxD

BEM,σxx

D Abaqus

(a) σDxx

-5

-4

-3

-2

-1

0

1 2 3 4 5

log(|

1-σ

/σre

f|)

log(dof)

τxyD

BEM,τxy

D Abaqus

(b) σDyy

Figure 16: Convergence of stress at domain point D vs. number of degrees of freedom (dof).

3.4 Laminated plate under shear load

Consider a laminated rectangular plate with base b = 200 mm (along x) and height h = 100 mm(along y), subjected to uniformly distributed shear load qy = −0.24 N/mm at x = b. The plateis clamped at x = 0. The laminate has 5 laminas, arranged in a [90/-90/0/-90/90] conguration.The thickness of each lamina is t = 0.3 mm. The laminated plate is modeled using its equivalentorthotropic properties [2, 6.4, Eq. (6.4)], i.e., Ex = 120 MPa, Ey = 60 MPa, Gxy = 7 MPa,νxy = 0.071. A typical mesh is shown in Figure 17. The element size is shown as h/Nel in Table 11.

Figure 17: Typical mesh used for Abaqus discretization. Laminated plate under shear load.

The results are compared with numerical results obtained using Abaqus at points A=(b,0), andB=(b,h). Comparison of displacements at points A and B is shown in Table 11. Convergence ofdisplacement at boundary points located at points A and B vs. degrees of freedom (dof) is shownin Figure 18. It can be seen that BEM converges to the displacements using less dof than Abaqus.



BEM Abaqus

N totel b/Nel dof UAx UBy N tot

el b/Nel dof UAx UBy

18 33.333 52 2.8786 12.5133 18 33.333 219 2.9083 12.290036 16.666 88 2.9099 12.5220 72 16.666 759 2.9284 12.454272 8.3333 160 2.9273 12.5436 288 8.3333 2811 2.9393 12.5254144 4.1666 304 2.9354 12.5553 1152 4.1666 10803 2.9445 12.5584288 2.0833 592 2.9392 12.5604 4608 2.0833 42339 2.9470 12.5746576 1.0416 1168 2.9411 12.5626 18432 1.0416 167619 2.9482 12.5826

-5

-4

-3

-2

-1

1 2 3 4 5 6

log(|

1-U

/Ure

f|)

log(dof)

UxA

BEM,Ux

A Abaqus

(a) UAx

-5

-4

-3

-2

-1

1 2 3 4 5 6

log(|

1-U

/Ure

f|)

log(dof)

UyB

BEM,Uy

B Abaqus

(b) UBy


4 Conclusions

The main contributions of this work are the analytical integration of singular kernels for planeorthotropic media, and a successful implementation into a BEM code. Numerical examples showthat the results obtained by the proposed BEM implementation are in good agreement with thoseobtained by FEM, demonstrating the validity and reliability of the proposed BEM formulation forthe analysis of elastic, 2D, orthotropic media. The analytical evaluation of the integral coecientsand the eciency of the implementation is highlighted. All equations are analytically integratedand expressed as a function of a few recurring integrals, contributed for the rst time in this work.The analytically integrated, HC interpolated BEM formulation converges much faster than a displa-cement-based FEM, for both displacements and stresses. Stresses always converge monotonically,while FEM sometimes displays oscillatory, slow convergence. The proposed model is able to re-present stress concentrations well. HC interpolation allows C1 continuity with negligible incrementof dof with respect to piecewise constant interpolation. The BEM formulation is mixed, providingdisplacements and tractions simultaneously and with the same order of approximation, which is sig-nicant for stress analysis. Further, the dimensionality of the model is reduced by one (from 2D to1D), which coupled with analytical integration and ecient interpolation leads to fast convergence


and economical solution with fewer dof than required by FEM.The proposed BEM formulation can be used to solve a variety of problems involving orthotropic

media. In the future, we wish to solve an orthotropic representative volume element that is themicro-model of a micro/macro model. Since the micro model must be solved for each Gauss pointat each iteration of the (nonlinear) macro model, computational cost is critical. The proposedformulation is clearly more economical that a FEM micro-model. Our intended applications is for aat panel. Therefore, we did not study the applicability and performance of the proposed methodto solve problems with curved geometries. Furthermore, our proposed application does not includebody forces.

Appendix

Coecients U(h)ij and T

(h)ij

Coecients U(h)ij , T

(h)ij are given next:

U(h)11 = D

[√λ1A

22G(1)

(h)0 −

√λ2A

21G(2)

(h)0

](45a)

U(h)12 = DA1A2

[A(2)

(h)0 −A(1)

(h)0

](45b)

U(h)21 = U

(h)12 (45c)

U(h)22 = −D

[A2

1√λ1G(1)

(h)0 − A2

2√λ2G(2)

(h)0

](45d)

For n1 = 0 and n2 = 1

T(h)11 = yD

[√λ2A1E(2)

(h)1 −

√λ1A2E(1)

(h)1

](46a)

T(h)12 = D

[√λ1A1E(1)

(h)1 −

√λ2A2E(2)

(h)1

](46b)

T(h)21 = D

[λ1

√λ1A2E(1)

(h)1 − λ2

√λ2A1E(2)

(h)1

](46c)

T(h)22 = yD

[√λ1A1E(1)

(h)1 −

√λ2A2E(2)

(h)1

](46d)

For n1 = 1 and n2 = 0

T(h)11 = yD

[√λ2A1E(2)

(h)1 −

√λ1A2E(1)

(h)1

](47a)

T(h)12 = D

[A1√λ1E(1)

(h)1 − A2√

λ2E(2)

(h)1

](47b)

T(h)21 = D

[√λ1A2E(1)

(h)1 −

√λ2A1E(2)

(h)1

](47c)

T(h)22 = yD

[√λ1A1E(1)

(h)1 −

√λ2A2E(2)

(h)1

](47d)


whereE(k)

(h)j = E(k)

(h+1)j − xE(k)

(h)j (48)

and n is the unit outward normal to the boundary took in the global reference system X,Y , son1 = nx and n2 = ny.

Coecients D(h)ijl and S

(h)ijl

Integral terms of the type D(h)ijl shown in (43), written as function of the recurring integrals, have

the following form

D(h)111 = D

[√λ2A1E(2)

(h)1 −

√λ1A2E(1)

(h)1

](49a)

D(h)122 = D

[λ1

√λ1A2E(1)

(h)1 − λ2

√λ2A1E(2)

(h)1

](49b)

D(h)121 = yD

[√λ2A1E(2)

(h)1 −

√λ1A2E(1)

(h)1

](49c)

D(h)112 = D

(h)121 (49d)

D(h)211 = yD

[A2√λ2E(2)

(h)1 − A1√

λ1E(1)

(h)1

](49e)

D(h)222 = yD

[√λ1A1E(1)

(h)1 −

√λ2A2E(2)

(h)1

](49f)

D(h)212 = D

[√λ1A1E(1)

(h)1 −

√λ2A2E(2)

(h)1

](49g)

D(h)212 = D

(h)221 (49h)

and integrals of the type S(h)ijl shown in (44), can be written for n1 = 0 and n2 = 1 as

S(h)111 = −2yD

[√λ2E(2)

(h)2 −

√λ1E(1)

(h)2

](50a)

S(h)112 = D

[√λ2E(2)

(h)1 −

√λ1E(1)

(h)1 − 2y2

(λ2E(2)

(h)2 − λ1E(2)

(h)1

)](50b)

S(h)121 = S

(h)211 = S

(h)112 (50c)

S(h)122 = 2yD

[λ2

√λ2E(2)

(h)2 − λ1

√λ1E(1)

(h)2

](50d)

S(h)212 = S

(h)221 = S

(h)122 (50e)

S(h)222 = D

[λ1

√λ1E(1)

(h)1 − λ2

√λ2E(2)

(h)1 + 2y2

(λ2

√λ2E(2)

(h)2 − λ1

√λ1E(2)

(h)1

)](50f)


and for n1 = 1 and n2 = 0, as

S(h)111 = D

[1√λ2E(2)

(h)1 − 1√

λ1E(1)

(h)1 − 2y2

(√λ2E(2)

(h)2 −

√λ1E(1)

(h)2

)](51a)

S(h)112 = −2yD

[√λ2E(2)

(h)2 −

√λ1E(1)

(h)2

](51b)

S(h)121 = S

(h)211 = S

(h)112 (51c)

S(h)122 = D

[√λ1E(1)

(h)1 −

√λ2E(2)

(h)1 + 2y2

(λ2

√λ2E(2)

(h)2 − λ1

√λ1E(1)

(h)2

)](51d)

S(h)212 = S

(h)221 = S

(h)122 (51e)

S(h)222 = 2yD

[λ2

√λ2E(2)

(h)2 − λ1

√λ1E(1)

(h)2

](51f)

with rk, λ1, λ2, Ak, zk, and D given in (5)(10).

Singular and near singular integrals

The calculation of the integrals on the boundary (14) brings to some integration problems, absentin the calculation of them in the domain. When the source point ξ is coincident with the eld pointx, the kernels of the integrals show some singularities.

The kernel function u∗ (ξ, x) have two weak singularities, O(ln zk) and O(arctan(r2/

√λkr1)

).

The kernel function t∗ (ξ, x) has a strong singularity O(1/z2k) [63].

The weak singularities disappear when we integrate the integrals on the boundary. The strongsingularity requires the evaluation of the integrals as as Cauchy Principal Value (CPV). The eva-luation of Cauchy Principal Value (CPV) integrals is one of the typical aspects of the BoundaryElement Method (BEM). It is essentially due to the strong singularity shown by some kernel functi-ons appearing in some boundary integral equations.

In this Appendix, the distance between source and eld point rk is denoted with ε to emphasizethat it is an innitesimal quantity.

Integration of kernels with singularity O(1/z2k)

Integration of t11 and t22

From the integral equation (14) consider the kernel function tij . When the distance between thesource point ξ and the eld point x tends to 0, ε → 0, the integration of kernel tij shows a strongsingularity that necessitates evaluation of the integral as CPV.


Figure 19: Singular part of t∗11.

We illustrate the integration procedure using t11. The integration of t22 follows the same proce-dure. Let's consider ∫

Γt∗

T

11 u dΓ = limε→0

∫

Γ−Γε

t∗T

11 ud(Γ− Γε)︸︷︷︸IA

+

∫Γε

t∗T

11 udΓε︸︷︷︸IB

where ε is distance between the source and eld point, where

IA = limε→0

∫Γ−Γε

t∗T

11 u d(Γ− Γε) =

∫Γt∗

T

11 udΓ

The term IA is part of the system (14), on the right side. This term has no singular contributionbecause it is far from the semi-circle in Fig (19). Next,

IB =

∫Γε

t∗T

11 udΓε = IB1 + IB2

can be decomposed as follows

IB = limε→0

∫Γε

t∗T

11 u dΓε = limε→0

∫Γε

D

[√λ2A1

z22

−√λ1A2

z21

](r1n1 + r2n2)

dΓε

where

IB1 = limε→0

∫Γε

t∗T


∫Γε

D

[√λ2A1

z22

](r1n1 + r2n2)

dΓε


and

IB2 = limε→0

∫Γε

t∗T

11 u dΓε = − limε→0

∫Γε

D

[√λ1A2

z21

](r1n1 + r2n2)

dΓε

Using polar coordinates x = ζ1 + ε cos θ

y = ζ2 + ε sin θ

where

ε =√

(x− ξ1)2 + (y − ξ2)2

r,1 =∂r1

∂x=x− ξ1

ε= cos θ −→ r1 = ε cos θ

r,2 =∂r2

∂y=y − ξ2

ε= sin θ −→ r2 = ε sin θ

n = [nx, ny] = [cos θ, sin θ]

dΓ = εdθ

zi =√λ1,2(x− ξ1)2 + (y − ξ2)2 =

√λ1,2(ε cos θ)2 + (ε sin θ)2

Then, εx− ξ1

ε= ε cos θ and taking in account (5), x− ξ1 = r1 and y − ξ2 = r2. Therefore,

IB1 =

= D√λ2A1 lim

ε→0

∫ π

0

cos θ2 + sin θ2

ε2 [1 + cos2 θ(λ2 − 1)]u ε2dθ = D

√λ2A1

1√λ2π =

A1 u

2(λ1 − λ2)S22

IB2 =

= −D√λ1A2 lim

ε→0

∫ π

0

cos θ2 + sin θ2

ε2 [1 + cos2 θ(λ1 − 1)]u ε2dθ = −D

√λ1A2

1√λ1π = − A2 u

2(λ1 − λ2)S22

Finally,

IB = IB1 + IB2 =A1 −A2

2(λ1 − λ2)S22u = c11

Therefore, the value of c11 for (14) is

c11 =A1 −A2

2(λ1 − λ2)S22= c22

Similarly, the term c22 results from integration of t22, using the same procedure described above.


Integration of t12 and t21

For integration of t12 and t21 we develop the following procedure

t12 = D

(√λ1A1

z21

−√λ2A2

z22

)r1n2 −

(√λ1

λ1

A1

z21

−√λ2

λ2

A2

z22

)r2n1

t12 = D

√λ1A1

(1

z21

r1n2 −1

λ1

1

z21

r2n1

)−√λ2A2

(1

z22

r1n2 −1

λ2

1

z22

r2n1

)

∫Γt∗

T

12 u dΓ = limε→0

∫

Γ−Γε

t∗T

12 ud(Γ− Γε)︸︷︷︸IA

+

∫Γε

t∗T

12 udΓε︸︷︷︸IB

where ε is distance between the source and eld point. Let's consider

IA = limε→0

∫Γ−Γε

t∗T

12 u d(Γ− Γε) =

∫Γt∗

T

12 udΓ

The term IA is part of RHS of (14). This term has no singular contribution because it is awayfrom the semi-circle in Fig. (19). Next,

IB =

∫Γε

t∗T

12 udΓε = IB1 + IB2

can be decomposed as follows

IB = limε→0

∫Γε

t∗T

12 u dΓε =

= limε→0

∫Γε

D

√λ1A1

(1

z21

r1n2 −1

λ1

1

z21

r2n1

)−√λ2A2

(1

z22

r1n2 −1

λ2

1

z22

r2n1

)dΓε

where

IB1 = limε→0

∫Γε

t∗T


∫Γε

D

√λ1A1

(1

z21

r1n2 −1

λ1

1

z21

r2n1

)dΓε

and

IB2 = limε→0

∫Γε

t∗T


∫Γε

−D√

λ2A2

(1

z22

r1n2 −1

λ2

1

z22

r2n1

)dΓε

Using polar coordinates

IB1 =

= limε→0

∫ π

0D

√λ1A1

(r1n2 −

1

λ1r2n1

)r

z21

udθ = −D√λ1A1 lim

ε→0

∫ π

0

1

z21

(1

λ1r2n1 − r1n2

)εudθ

IB2 =

= limε→0

∫ π

0−D

√λ2A2

(r1n2 −

1

λ2r2n1

)r

z22

udθ = D√λ2A2 lim

ε→0

∫ π

0

1

z22

(1

λ2r2n1 − r1n2

)εudθ


IB1 = −D√λ1A1 lim

ε→0

∫ π

0

1

z21

(1

λ1cos θ sin θ − cos θ sin θ

)ε2udθ =

= −D√λ1A1 lim

ε→0

∫ π

0

(1

λ1− 1

)cos θ sin θ

ε2 [1 + cos2 θ (λ1 − 1)]

ε2udθ = 0

IB2 = D√λ2A2 lim

ε→0

∫ π

0

1

z22

(1

λ2cos θ sin θ − cos θ sin θ

)ε2udθ =

= D√λ2A2 lim

ε→0

∫ π

0

(1

λ2− 1

)cos θ sin θ

ε2 [1 + cos2 θ (λ2 − 1)]

ε2udθ = 0

results in

c12 = c21 = 0

Integration of kernels with singularity O(ln zk)

Integration of u11 and u22

For integration of u12 and u21 we use the following procedure

∫Γu∗

T

11 t dΓ = limε→0

∫

Γ−Γε

u∗T

11 td(Γ− Γε)︸︷︷︸IA

+

∫Γε

u∗T

11 tdΓε︸︷︷︸IB

where ε is distance between the source and eld point. Expanding the two integrals on the RHS,we have

IA = limε→0

∫Γ−Γε

u∗T

11 t d(Γ− Γε) =

∫Γu∗

T

11 tdΓ

and

IB = limε→0

∫Γε

u∗T

11 t dΓε = limε→0

∫Γε

D[√

λ1A22 ln z1 −

√λ2A

21 ln z2

]tdΓε

or

IB = limε→0

∫Γε

u∗T


∫Γε

D[√

λ1A22 ln z1

]tdΓε − lim

ε→0

∫Γε

D[√

λ2A21 ln z2

]tdΓε =

= D√λ1A

22 limε→0

∫ π

0ε2ln

[√1 + cos2 θ (λ1 − 1)

]tdθ−D

√λ2A

21 limε→0

∫ π

0ε2ln

[√1 + cos2 θ (λ2 − 1)

]tdθ = 0

Integration of kernels with singularity O(arctan(r2/

√λkr1)

)Integration of u12 = u21

The integration procedure for u12 is as follows

∫Γu∗

T

12 t dΓ = limε→0

∫

Γ−Γε

u∗T

12 td(Γ− Γε)︸︷︷︸IA

+

∫Γε

u∗T

12 tdΓε︸︷︷︸IB


where ε is distance between the source and eld point. Expanding the two integrals on the RHS,we have

IA = limε→0

∫Γ−Γε

u∗T

12 t d(Γ− Γε) =

∫Γu∗

T

12 tdΓ

and

IB = limε→0

∫Γε

u∗T


∫Γε

DA1A2

[arctan

(r2√λ2r1

)− arctan

(r2√λ1r1

)]tdΓε

or

IB = limε→0

∫ π

0u∗

T

12 t dθ = limε→0

∫ π

0DA1A2

[arctan

(sin θ√λ2 cos θ

)− arctan

(sin θ√λ1 cos θ

)]tεdθ =

= limε→0

∫ π

0DA1A2

[(θ√λ2

)−(

θ√λ1

)]tεdθ = 0

Acknowledgements

The authors wish to acknowledge the nancial support from the RISPEISE cooperation project.

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analytic integration of singular kernels for boundary

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